One of the priority issues of the modern higher education is the development of international programs of student and teacher mobility. Such programs enable Russian higher education institutions to demonstrate their competitiveness at the market of educational services not only in their country but also elsewhere in the world. In many developed countries, universities are engaged not only in training; moreover, they are centers for advanced scientific researches. Modern Russian universities have chosen a similar development strategy; they tend to move away gradually from the Soviet model, according to which most scientific researches were to be conducted by research institutes which were not affiliated with higher education institutions. The article examines the experience of cooperation between the Peoples' Friendship University of Russia (RUDN, Russia) and the University of the Balearic Islands (UIB, Spain) in training highly qualified academic specialists. The Peoples' Friendship University of Russia is one of the modern Russian universities seeking to become a full-fledged research institution. In this context, particular attention is given to the training of international-level academic specialists. The key problem in this area is the inconsistency of academic degrees in the Russian and Western education systems. In the West, after receiving Master's degree one can enter a PhD program, whereas in Russia there remains a two-stage system of postgraduate education: Candidate and Doctor of Sciences (PhD degree and doctor's degree). This complicates the cooperation between Russian universities and their foreign partners. One of the possible solutions to the problem is the signing of a special agreement between universities, which was developed by representatives of RUDN and UIB. The agreement was signed in 2015; and in October 2018, under the agreement there took place the first successful PhD defense. The article analyzes the positive aspects and difficulties of implementing this agreement. Furthermore, there are outlined the further prospects for cooperation in this area. The program can also be a positive example for other Russian universities and their foreign partners.